Commutativity Theorems for *-Prime Rings with Differential Identities on Jordan Ideals

In this paper we explore commutativity of -prime rings in which derivations satisfy certain differential identities on Jordan ideals. Furthermore, examples are given to demonstrate that our results cannot be extended to semiprime rings. 1. Introduction Throughout this paper, will represent an associative ring with center . is -torsion free if yields . We recall that is prime if implies or . A ring with involution is -prime if yields or . It is easy to check that a -prime ring is semiprime. Moreover, every prime ring having an involution is -prime but the converse does not hold, in general. For example, if denotes the opposite ring of a prime ring , then equipped with the exchange involution , defined by , is -prime but not prime. This example shows that every prime ring can be injected in a -prime ring and from this point of view -prime rings constitute a more general class of prime rings. In all that follows will denote the set of symmetric or skew-symmetric elements of . For and . An additive subgroup of is a Jordan ideal if for all and . Moreover, if , then is called a -Jordan ideal. We will use without explicit mention the fact that if is a Jordan ideal of , then and [1, Lemma 1]. Moreover, From [2] we have ,？ and for all . A mapping is called strong commutativity preserving on a subset of if for all . An additive mapping is called a derivation if holds for all pairs . Recently, many authors have obtained commutativity theorems for -prime (prime) rings admitting derivation, generalized derivation, and left multiplier (see [3–8]). In this paper, we will explore the commutativity of -prime rings equipped with derivations satisfying certain differential identities on Jordan ideals. 2. Differential Identities with Commutator We will make some use of the following well-known results. Remarks 2.1. Let be a -torsion free -prime ring and a nonzero -Jordan ideal.(1) (see [6, Lemma 2]) If , then or .(2) (see [6, Lemma 3]) If , then .(3) (see [7, Lemma 3]) If , then is commutative.(4) (see [9], Lemma 3]) If is a derivation such that for all , then . We leave the proofs of the following two easy facts to the reader.(5) If , then . In particular, if or , then .(6) If admits a derivation such that , then . Lemma 2.2. Let be a 2-torsion free -prime ring and a nonzero -Jordan ideal. If admits a nonzero derivation such that for all , then is commutative. Proof. First suppose that . From it follows that Substituting for in (2.1), where , we obtain which leads to For , (2.2) together with Remarks 2.1(1) forces , in which case , or . Since , in both the cases we arrive